I rented a car and declined the insurance coverage offered by the rental company. 

There is some cosmetic damage to the car... the result of coming
around corner of a parking garage too close. There are scrapes and
some scratches.

When returning the rental car, what is the process for assessing and
determining the cost to repair the damages? I have rental insurance
covered through the credit card which I plan to use to cover the costs
but I'm just curious about what to expect at the rental company when I
return the car.

I had the same exact problem as you, from a hotel parking garage in
California a few years ago.  My solution was to take the car to the
local auto parts store where I bought a buffing compound and cloth and
spent about an hour removing the scratches.  This $10 / 1 hour
solution was much better than having to deal with the potential
problems from the rental car company.